Universal oscillations in counting statistics

Abstract

Noise is a result of stochastic processes that originate from quantum or classical sources. Higher-order cumulants of the probability distribution underlying the stochastic events are believed to contain details that characterize the correlations within a given noise source and its interaction with the environment, but they are often difficult to measure. Here we report measurements of the transient cumulants ͗͗n m ͘͘ of the number n of passed charges to very high orders (up to m ‫؍‬ 15) for electron transport through a quantum dot. For large m, the cumulants display striking oscillations as functions of measurement time with magnitudes that grow factorially with m. Using mathematical properties of high-order derivatives in the complex plane we show that the oscillations of the cumulants in fact constitute a universal phenomenon, appearing as functions of almost any parameter, including time in the transient regime. These ubiquitous oscillations and the factorial growth are system-independent and our theory provides a unified interpretation of previous theoretical studies of high-order cumulants as well as our new experimental data.cumulants ͉ distributions ͉ electron transport ͉ noise and fluctuations C ounting statistics concerns the probability distribution P n of the number n of random events that occur during a certain time span t. One example is the number of electrons that tunnel through a nanoscopic system (1-4). The first cumulant of the distribution is the mean of n, ͗͗n͘͘ ϭ ͗n͘, the second is the variance, ͗͗n 2 ͘͘ ϭ ͗n 2 ͘ Ϫ ͗n͘ 2 , the third is the skewness, ͗͗n 3 ͘͘ ϭ ͗(n Ϫ ͗n͘) 3 ͘. With increasing order the cumulants are expected to contain more and more detailed information on the microscopic correlations that determine the stochastic process. In general, the cumulants ͗͗n m ͘͘ ϭ S (m) (z ϭ 0) are defined as the mth derivative with respect to the counting field z of the cumulant generating function (CGF) S(z) ϭ ln͚ n P n e nz . Recently, theoretical studies of a number of different systems have found that the high-order cumulants oscillate as functions of certain parameters (5-9), however, no systematic explanation of this phenomenon has so far been given. Examples include oscillations of the high-order cumulants of transport through a Mach-Zender interferometer as functions of the Aharonov-Bohm flux (6), and in transport through a double quantum dot as functions of the energy dealignment between the two quantum dots (8). As we shall demonstrate, oscillations of the high-order cumulants in fact constitute a universal phenomenon which is to be expected in a large class of stochastic processes, independently of the microscopic details. Inspired by recent ideas of M. V. Berry for the behavior of high-order derivates of complex functions (10), we show that the high-order cumulants for a large variety of stochastic processes become oscillatory functions of basically any parameter, including time in the transient regime. We develop the theory underlying this surprising phenomenon and present the first...

High-order cumulants in the counting statistics of asymmetric quantum dots

“…For the 12th order values up to 145 are reached and the amplitude of the 15th cumulant already exceeds 10 000. This factorial increase was shown already for oscillations as function of time 12 and appears here again, showing the universal nature of the oscillations.…”

“…14 The theoretical treatment has shown that these oscillations are a universally expected phenomenon for most physical systems and that in QDs they should show up prominently as function of the barrier asymmetry. 12,16 In this paper, we present the experimental verification of the predicted oscillations in the higher cumulants as function of barrier asymmetry in a QD system. To this aim we employ real time single electron counting with a high bandwidth detector.…”

“…Recently it was found experimentally that the high-order cumulants of QD transport oscillate as function of the counting time interval. 12 In the same paper, it is stated that such oscillations are expected universally for nearly any system when changing a relevant parameter, e.g., the symmetry of the tunneling rates in a QD system. 12 For our experimental study of this predicted symmetry dependence of high-order cumulants we varied the asymmetry parameter a and measured for each a about one million tunneling events.…”

“…This correspondence between the different parameters nicely demonstrates that the occurrence and the strength of these oscillations are of universal nature. 12 It is also interesting to examine the behavior at long times or toward the case of symmetric rates, a = 0. For long times the dependence on ⌫ S ϫ t 0 becomes weak and in the long time limit the cumulants will become sole function of the asymmetry.…”

Waiting time distribution of quantum electronic transport in the transient regime

“…[5][6][7][8] Experimentally, high order transient cumulants of charge passing through a quantum point contact have been measured up to 15th cumulant and universal oscillations were found in counting statistics. 9 Theoretically, finite frequency FCS has been studied using quantum master equation and scattering matrix approach. [10][11][12][13] Less attention has been paid on FCS of transient transferred charge.…”